Risk-neutral measure of financial derivatives pricing in a very key concept. For all the well-known Black Scholes pricing formula can be drawn from the two methods, one of which is through stock options and construct a risk-free investment portfolio, to the introduction of the option price through a combination of construction and real payoff consistency of the underlying risk-free satisfied by a partial differential equation to obtain by solving partial differential equations of the option price. The other one is risk-neutral measure with very relevant martingale method, by constructing a risk-neutral measure, and then the options in the future payoff expectations by seeking risk-neutral measure to get the price of the option.
Then the risk neutral measure in the end what is it? From the perspective of more stringent, the price of all assets is carried out on the products discounted by the market risk neutral measure is a martingale. The martingale is a stochastic process refers to his time in the value of any future expectations, equal to the value now. That is, if the product is the price of assets martingale, then people will not be able to predict future price movements. With a relatively concise words, the risk-neutral measure, that is, the price of an asset, the asset will be a payoff risk-free rate of units when an event occurs, and the payoff at the other events is 0. We said the assets Arrow assets.
And even say, not very able to understand, so we expect to be explained by risk-neutral measure. Suppose a property, there are n events that may arise, then for different events such assets have n different payoff, its value is equal to Y (n) a risk-free rate of payoff, how do we determine the price of the assets it ? Copy the payoff method can be used, for each of the different events that we are using Y (n) th Arrow assets (if the incident payoff for the risk-free rate of one unit, other events payoff 0) to copy, then the final The result is:
SUM (X number of price) = SUM (Y (n) * p (n))
Where p (n) is the n-th Arrow to copy the price of assets, to meet the requirements of his measure, it can be called risk-neutral measure, or risk-neutral probability. The above equation is just the risk of the Y value obtained by taking a desired neutral measure, while understood, P n is a probability of occurrence, also called neutral measure risk / probability.
Risk-neutral measure in real life the actual probability measure is equivalent to that event risk neutral measure equal to 0, the actual probability measure also is 0, and 0 in the event greater than the risk-neutral measure, the actual probability measure is also greater than zero. In the derivation of the above risk-neutral measure, we use the Arrow asset pricing, its future payoff is based on risk-free rate in units of measure, which is known as the risk-free interest rate numeraie, and we can be varied by use of numeraie other assets as the payoff of denominations, but the asset must be a tradable asset and the price of his process must always be greater than zero.
For the risk-neutral measure, there are two very classic and useful theorems, the first one is the theorem of asset pricing basis, consists of two parts. The first part says that if there is a risk-neutral measure on a market, then the market is no arbitrage, that is, when using the Black Scholes pricing formula, we have assumed that the market is arbitrage-free, so there is no BS deviates from the formula developed asset prices, in theory, there will be some arbitrage opportunities, but in fact there is a BS implied volatility of uncertainty, this part of the deep, in the future we will gradually introduce. The second part says if one does not exist and there is only one measure of risk-neutral arbitrage on the market, then the market is complete, that is the price of any asset can be copied.
While the second is very classic theorems Girsanov theorem for:
. \ [The Z (T) = exp \ bmatrix the begin {} - \ ^ int T_0 \ Theta (U) dW (U) - \ FRAC. 1 {{} 2} \ int ^ t_0 || \ Theta (u) || ^ 2du \ end {bmatrix} \]
\ [\ Widetilde {W} (
t) = W (t) + \ int ^ t_0 \ Theta (u) du \] where Theta is adapted random process multidimensional, W is a Brownian motion, we define:
\ [\ widetilde {P} (A) = \ int_A Z (w) dP (w) for \ quad all \ quad A \ in F \]
Then Z in the desired measure is 1, and a Brownian motion is defined above. We can easily observe the definition of the measure is the risk-neutral measure, and the actual probability measure equivalent to P, and the new definition of Brownian motion is defined under the risk-neutral measure of Brownian motion. Through the above theorems, we found that the risk-neutral measure relative to the actual probability measure, change only measure of expectations, and the second moment of volatility measure has not changed.
Risk-neutral pricing measure applied in a very wide range of derivatives, so you want to be not only heard, but also detailed understanding of them, but only back down BS formula, can not be truly understood.